Matthew Wells,
Summer 2001
GVSU SURP (Summer Undergraduate Research Project) Grant
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Matt did a wide variety of work that actually occurred over 2+ years,
starting in summer 2000. He devoted a sizable amount of his time to investigating the
ratio vectors of polynomials. The so-called "ratio vector"
comes from a result called Peyser's Interval Theorem [1967]: If p is a polynomial with distinct real zeros r1,r2,...,rn,
and critical numbers c1,c2,...,cn-1,
then the following inequalities hold, for i = 1,...,n-1:
The inner term in this inequality is called the "ith ratio", and the collection of all n - 1 of these is the "ratio vector". (This result was independently re-discovered by P. Andrews in 1995.) |
Matt and I (mostly Matt) proved multiple results
about the ratio vector for polynomials with equally spaced zeros, as well as
for the famous Chebyshev polynomials. A summary of some of his work is
found in this paper, which was published
in the Missouri Journal of Mathematical Sciences early in 2007. He also made a very interesting conjecture about
PIPCIRs, Polynomials whose Inflection Points Coincide with their Interior
Zeros. That conjecture led to another paper,
which appeared in the Pi Mu Epsilon Journal in Fall 2006. He gave a large number of talks on
his work, including a presentation at MathFest 2001 in Madison, WI.
Matt is now a doctoral candidate in mathematics at the
University of Kentucky.
Jennifer Miller
(Bucknell University) and Benjamin Vugteveen (GVSU),
Summer 2003, GVSU REU, NSF grant
DMS-0137264
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The three of us worked together on two main problems for 8 weeks. The first (at
which we were almost, but not quite, successful) was to extend Matt Wells'
work on the ratio vectors of Chebyshev polynomials of the first kind to the
other 3 kinds of Chebyshev polynomials. The 2nd, 3rd, and 4th kind
Chebyshev polynomials exhibit many of the same patterns that those of the
first kind do, but for reasons not worth going into here, they prove to be
substantially harder to work with.
While working on ideas related to polynomial root-dragging and ratio
vectors, Jen, Ben, and I hit upon a perspective of how root-dragging affects
the supremum norm of a polynomial. We used this concept to ultimately
prove an inequality that shows the largest (in absolute value) that certain
polynomials with zeros only in the complex unit disk can be. The
result reveals some interesting links to the famous Bernstein polynomials, and
adds perspective to known properties of Chebyshev polynomials. We also
have
a paper that appeared in May 2006 in the College
Math Journal. Jen and Ben gave several talks to the REU, spoke at the
Michigan REU conference, also spoke at MathFest 2003 in Boulder, CO.
Jen is pursuing graduate study at the University of Illinois. Ben
works for Epic Systems in Madison, WI.
John Skukalek, Winter
2004
GVSU Senior Thesis
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Early on in having John as a student, I discovered that he was as
fond of infinite series as I was. When I shared the fact of Euler's
solution to the Basel
Problem, John came up and asked "How do you prove that?"
That led him to read Dunham's great book (Euler:
The Master of us All), and from there, well, John went on to be one of
our finest math majors ever. He graduated in April 2004 (winning the
university's prized Niemeyer Award) and is currently a PhD candidate at
Penn State University.
For his undergraduate thesis, John investigated about 10 different solutions to the Basel problem using such diverse techniques as Fourier analysis and the calculus of residues from complex variables. |
Further,
he read some challenging papers in the field of analytic number theory that
use multiple integrals to establish proofs of irrationality of certain numbers, including the relatively recent result that the sum of
the reciprocals of the cubes is irrational. He wrote a beautiful paper
(available here in .pdf format) that
summarizes all of this work, and gave a stellar seminar talk to the math
department faculty.
Justin From
(Central College, IA) [left, below] and Samuel Kolins (Bowdoin
College) [right],
Summer 2005, GVSU REU, NSF grant
DMS-0451254
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Sam, Justin, and I worked together for 8 weeks on problems in the geometry of polynomials. The first problem, which was related to an open conjecture of R. Robinson, involved the span of a polynomial with all real zeros. The span of such a function is the distance between its greatest and least roots. Sam and Justin developed several theorems and a clever new argument to show that among all polynomials with span 2, there is one monic polynomial with roots at -1 and 1 whose derivatives each have the smallest possible span, and that function is
The
heart of their work is their Polynomial Root Squeezing Theorem. This
result shows that if two roots of a polynomial with all real zeros are moved
towards each other, then the "outer" critical numbers of the
function move inward as well. We
wrote a paper about our work that has been accepted for publication by Mathematics
Magazine and will appear in early 2008. In
addition, Sam and Justin began learning about the Hausdorff geometry of
polynomials, and made some significant investigations into a problem related
to the Sendov Conjecture, including complete resolution of two different situations involving
cubic polynomials that were highly nontrivial.
These two gentlemen
were outstanding students and a genuine pleasure to supervise; their work
was independent, voluminous, and creative. The technical
report they developed over the course of the program numbers over 60 pages
in length and demonstrates the impressive range of mathematics they learned
and discovered, as well as their great communication skills. Each gave
a wonderful talk at Mathfest 2005 in Albuquerque, NM; Sam was chosen to
receive one of 8 prizes from among 90 undergraduate student talks given in
MAA sponsored sessions.
Justin has plans to attend law school after spending time serving Teach for
America. Sam is attending Cornell
University to study for his PhD in mathematics.
Future projects:
There are many possibilities for continued research studying the set of
critical numbers relative to the set of zeros of polynomials. A particularly exciting one is related to the Sendov
Conjecture, a result that focuses on complex polynomials, all of whose roots
lie in the complex unit disk. Posed by Blagovest Sendov in 1962, the
conjecture essentially states that, given any polynomial having all its
zeros in the complex unit disk, if we choose any particular root of the
function, there must be at least one critical number within 1 unit distance
from that particular root. Sounds simple, yes? Alas, the general
case turns out to be amazingly difficult.
Mathematicians resolved the conjecture in a couple of cases fairly quickly
-- not surprisingly for polynomials of degree 3 and 4, and a couple other
basic situations. Then there was about a 20 year interlude where no
progress was made. Since 1995 or so, a significant amount of research
has been done on this problem, and now it is currently known that the Sendov
Conjecture is true for all polynomials having up to 8 distinct roots in the
unit disk. The research in this area has led to a number of other
interesting problems, some of which are accessible to undergraduates.
In the sense of the Hausdorff metric, the Sendov conjecture is saying that
the largest possible deviation between the set of zeros and the set of
critical numbers of a polynomial (with all zeros in the unit disk) is
1. There are several other related conjectures, as well as a range of
concepts that deal with related sets -- like the convex hull, circumself
disk, gravity disk, and more on the sets of zeros and critical
numbers. One can also turn the conjecture around a bit and ask (in
certain circumstances) what the smallest possible deviation between
these sets is. In several instances, the case where the polynomial is
real and has all real distinct roots remains unresolved.
In the summer 2008, I will be one of four mentors in the GVSU
REU program. I look forward to supervising two new students and
seeing the mathematics that we can discover together; the topic of our
investigations will be something related to the problems on the geometry of
polynomials that have been described above. If you are a prospective REU student, or are
simply interested in learning more, please don't hesitate to contact me at
boelkinm at gvsu dot edu.